2. Q.1: REPRESENT REAL NUMBERS THROUGH VEN DIAGRAM
REAL
NUMBERS
IRRATIONAL NUMBERS
RATIONAL NUMBERS
INTEGERS AND FRACTIONS
WHOLE NUMBERS
(2)/(2)
9.4545...
-
3
0
1
2
5
7 826
157239
2
15
1.243487....
A.1:
3. Q.2: Fun Facts about Polynomials
• If P(x) is a polynomial of degree n in one variable, with complex
coefficients, then it factors into linear terms: P(x) = C(x − r1)(x − r2)· · ·(x
− rn) where r1, . . . , rn are the roots of P(x) – these are often complex
numbers.
• If P(x) is a polynomial in one variable, and P(a) = 0, then P(x) is
divisible by x − a: that is, there exists Q(x) such that P(x) = (x − a)Q(x).
• A polynomial of degree n in one variable has at most n different roots.
• If P(x) is a polynomial of degree n in one variable with real coefficients,
then P(x) factors as a product of linear and quadratic terms with real
coefficients.
• If P(x) and D(x) are polynomials in one variable, then we can write P(x)
= Q(x)D(x) + R(x) where the degree of R(x) is less than the degree of
D(x). Here, R(x) stands for the remainder of P(x) when divided by D(x).
5. Q.4: Fun Facts about Polynomials
• Polynomials appear in many areas of mathematics and science. For
example, they are used to form polynomial equations, which encode a
wide range of problems, from elementary word problems to complicated
scientific problems; they are used to define polynomial functions.
• The word polynomial joins two diverse roots: the Greek poly, meaning
"many", and the Latin nomen, or name. It was derived from the term
binomial by replacing the Latin root bi- with the Greek poly-. The word
polynomial was first used in the 17th century.
• A polynomial in one indeterminate is called a univariate polynomial, a
polynomial in more than one indeterminate is called a multivariate
polynomial. A polynomial with two indeterminates is called a bivariate
polynomial.These notions refer more to the kind of polynomials one is
generally working with than to individual polynomials.
6. Q.5: REAL-LIFE EQUATIONS
Q.1: Five years ago, Navya was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu.
How old are Navya and Sonu?
A.1: Let us assume, present age of Nuri is x and present age of Sonu is y.
According to the given condition, we can write as;
x – 5 = 3(y – 5)
x – 3y = -10…………………………………..(1)
Now, x + 10 = 2(y +10)
x – 2y = 10…………………………………….(2)
Subtract eq. 1 from 2, to get,
y = 20 ………………………………………….(3)
Substituting the value of y in eq.1, we get,
x – 3.20 = -10
x – 60 = -10
x = 50
Therefore,
Age of Nuri is 50 years
Age of Sonu is 20 years.
7. Q.5: REAL-LIFE EQUATIONS
Q.1: Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her Rs.50 and Rs.100
notes only. Meena got 25 notes in all. Find how many notes of Rs.50 and Rs.100 she received.
A.1: Let the number of Rs.50 notes be A and the number of Rs.100 notes be B
According to the given information,
A + B = 25 ……………………………………………………………………….. (i)
50A + 100B = 2000 ………………………………………………………………(ii)
When equation (i) is multiplied with (ii) we get,
50A + 50B = 1250 …………………………………………………………………..(iii)
Subtracting the equation (iii) from the equation (ii) we get,
50B = 750
B = 15
Substituting in the equation (i) we get,
A = 10
Hence, Meena has 10 notes of Rs.50 and 15 notes of Rs.100.
8. Q.5: REAL-LIFE EQUATIONS
Q.1: A lending library has a fixed charge for the first three days and an additional charge for each day
thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs.21 for the book she
kept for five days. Find the fixed charge and the charge for each extra day.
A.1: Let the fixed charge for the first three days be Rs.A and the charge for each day extra be Rs.B.
According to the information given,
A + 4B = 27 …………………………………….…………………………. (i)
A + 2B = 21 ……………………………………………………………….. (ii)
When equation (ii) is subtracted from equation (i) we get,
2B = 6
B = 3 …………………………………………………………………………(iii)
Substituting B = 3 in equation (i) we get,
A + 12 = 27
A = 15
Hence, the fixed charge is Rs.15
And the Charge per day is Rs.3
9. Q.5: BIRTHDAY PROBABLITIES
Q.1: A and B are friends . What is the probability that both will have birthdays
a) on the same days. B ) on different days [ 1 year = 365 days ].
A.1: No. of days in a year = 365
Since there are two birthdays(A and B) in a year,
n(S) = 365 x 365 -----------(1)
b) Let E be the event that A and B have different birth days.
But of 365 days A can have birthday on any day and hence B can have birthday on (365 - 1) i.e. 364
days.
∴ n(E) = 365 × 364
Now, P(E) = n(E) / n(S) = (365x364) / (365x365) = 364 / 365.
a) Let F be the event that both of them have their birth days on same day.
E and F are complementary events.
∴ P(F) = 1 - P(E)
Using value of P(A) from (i),
P(F) = 1 – 364 / 365 = 1 /365.
10. Q.7: SHAPES AND THE SURFACE FORMULAE
Shape: Cube
Surface Area: 6a^2
Volume: a^3
Shape: Cuboid
Surface Area: 2(lb+bh+hl)
Volume: l * b * h
Shape: Cone
Surface Area: π r(r+l)
Volume: 1/3π r^2 h
Shape: Cylinder
Surface Area: 2 π r(r+h)
Volume: π r^2 h
Shape: Sphere
Surface Area: 4 π r^2
Volume: 4/3π r^3
11. Q.8: PRIMARY AND SECONDARY DATA
• Primary data are those that are collected for the first time. Secondary data
refer to those data that have already been collected by some other person.
These are original because these are collected by the investigator for the
first time.
• WEST BENGAL
- Parliamentary constituencies: 42
- Ruling Party: All India Trinamool Congress
• TAMIL NADU
- Parliamentary constituencies: 39
- Ruling Party: Dravida Munnetra Kazhagam
• KERALA
- Parliamentary constituencies: 20
- Ruling Party: Communist Party of India
This is a secondary data since its bein collected by some other
source.
